Finite Group Databasebeta

← #27294692

Subgroups — #27294692

Order 13568.

514 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133134C211yesmaximal, characteristic, complemented
2#20273053C211yesmaximal, characteristic, complemented
3#20272721C212yesmaximal, complemented
4#20272718C212yesmaximal, complemented
5C53×#2543C211yesmaximal, characteristic
6C2×#13133134—21no—
7#16810587—21no—
8C4×#11719917—21no—
9#16810392C2212yes—
10#16810349C2212yes—
11#16810347—21no—
12C2×#121×C53—21no—
13C2×#121×C53C411yescharacteristic
14C2×C4×D4×C53C2211yescharacteristic
15C53×#398C2211yescharacteristic
16C53×#397C411yescharacteristic
17C53×#394C2211yescharacteristic
18C53×#392C411yescharacteristic
19C53×#388—21no—
20C53×#386—21no—
21C53×#386—21no—
22C2×#95×C53C411yescharacteristic
23C2×#11719917—22no—
24C2×#11719917—41no—
25#13133134—24no—
26C22⋊#88322—22no—
27C22⋊#88322—41no—
28#13133128—41no—
29C4×#88322—41no—
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53C2×C411yescharacteristic
33C22×D4×C53D411yescharacteristic
34C23×C4×C53—21no—
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53D411yescharacteristic
37#121×C53—41no—
38#121×C53—41no—
39#121×C53—44no—
40C4×D4×C53—41no—
41C4×D4×C53—41no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53C2311yescharacteristic
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C42×C53D411yescharacteristic
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53—21no—
59C24×C53C22⋊C411yescharacteristic
60C24×C53C22⋊C411yescharacteristic
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—41no—
64C2×D4×C53—42no—
65C2×D4×C53—41no—
66C2×D4×C53—42no—
67C2×D4×C53—42no—
68C2×D4×C53—42no—
69C2×D4×C53—42no—
70C2×D4×C53—81no—
71C2×D4×C53—81no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53—81no—
82C22×C4×C53—81no—
83C22×C4×C53—81no—
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22×C411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—42no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—42no—
95C22⋊C4×C53—42no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—42no—
99C22⋊C4×C53—81no—
100C42×C53—21no—
101C42×C53—21no—
102#11719917—24no—
103#11719917—42no—
104#11719917—82no—
105C2×#88322—42no—
106C2×#88322—42no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53#10412yes—
124C23×C53C2×C22⋊C411yescharacteristic
125D4×C53—44no—
126D4×C53—84no—
127D4×C53—82no—
128D4×C53—84no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—22no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—41no—
136C2×C4×C53—42no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—81no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53—84no—
152C2×C4×C53#9911yescharacteristic
153C2×C4×C53#9911yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)12yes—
155#88322—44no—
156#88322—82no—
157#11783—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—22no—
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53C2×#9911yescharacteristic
178C22×C53#41811yescharacteristic
179C22×C53C2×#10411yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—22no—
183C4×C53—42no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—81no—
188C4×C53—81no—
189#2543D5311yescharacteristic
190C4×#100—531no—
191#1465—531no—
192#1346—532no—
193#1345—532no—
194C2×C53—21no—
195C2×C53—21no—
196C2×C53—42no—
197C2×C53—81no—
198C2×C53—161no—
199C2×C53#150711yescharacteristic
200C2×C53#212111yescharacteristic
201C2×C53#213211yescharacteristic
202C2×#121—21no—
203C2×#121Dic5311yescharacteristic
204C2×C4×D4C2×D5311yescharacteristic
205C2×#100—1061no—
206#407—1061no—
207C4×M16—1061no—
208#398C2×D5311yescharacteristic
209#397Dic5311yescharacteristic
210#394C2×D5311yescharacteristic
211#392Dic5311yescharacteristic
212#388—21no—
213#386—21no—
214#386—21no—
215C2×#95Dic5311yescharacteristic
216#343—532no—
217#324—532no—
218#323—1061no—
219C53#1178311yescharacteristic, complemented, Sylow
220C22×D4—21no—
221C22×D4—21no—
222C22×D4C2×Dic5311yescharacteristic
223C22×D4C22⋊D5311yescharacteristic
224C23×C4—21no—
225C23×C4C2×Dic5311yescharacteristic
226C23×C4C22⋊D5311yescharacteristic
227C2×M16—1062no—
228C2×M16—2121no—
229#121—44no—
230#121—41no—
231#121—41no—
232C4×D4—41no—
233C4×D4—41no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C2×Dic5311yescharacteristic
236C2×C4⋊C4C22×D5311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4C2×Dic5311yescharacteristic
243C2×C22⋊C4C22⋊D5311yescharacteristic
244C2×C42C22⋊D5311yescharacteristic
245#100—1064no—
246C22⋊C8—1062no—
247C22⋊C8—2121no—
248#97—2121no—
249C4×C8—2121no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95C2×Dic5311yescharacteristic
254#95C2×Dic5311yescharacteristic
255C24—21no—
256C24C22⋊Dic5311yescharacteristic
257C24C22⋊Dic5311yescharacteristic
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—81no—
268C2×D4—81no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4C22⋊Dic5311yescharacteristic
282C22×C4C22⋊Dic5311yescharacteristic
283C22×C4C22×Dic5311yescharacteristic
284C22×C4C2×C22⋊D5311yescharacteristic
285C22×C4C2×C22⋊D5311yescharacteristic
286M16—1064no—
287M16—4242no—
288M16—2122no—
289C2×C8—2122no—
290C2×C8—2122no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—81no—
302C42—21no—
303C42—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23#1313314412yes—
321C23C2×C22⋊Dic5311yescharacteristic
322D4—44no—
323D4—84no—
324D4—82no—
325D4—84no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—84no—
349C2×C4#1313313211yescharacteristic
350C2×C4#1313313211yescharacteristic
351C2×C4#1313324312yes—
352C8—4242no—
353C8—2124no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22C2×#1313313211yescharacteristic
374C22#1681062811yescharacteristic
375C22C2×#1313314411yescharacteristic
376C4—22no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C2—21no—
386C2—21no—
387C2—42no—
388C2—81no—
389C2—161no—
390C2#2027323111yescharacteristic
391C2#2027636111yescharacteristic
392C2#2027640011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
Fitting, prime radicalC53×#2543C2
socleC22×C53C2×#99
cosocleC2×C4⋊C4×C53C23
hypercenter#2543D53
hypocenterC53#11783
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2543

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C4×#13133134C2semidirect
2. #20273053C2semidirect
3. #20272721C2semidirect
4. #20272718C2semidirect
219. C53#11783semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×#131331342
#202730532
#202727212
#202727182
C53×#25432
#1178353